How to Solve Absolute Value Equations and Inequalities

Here's a question that trips up a lot of people: why does $|x| = 5$ have two answers?

Nothing about the equation looks like it should. It's just one variable, one number, one equals sign. But the answer is $x = 5$ or $x = -5$, and if you don't know why, the whole topic feels like a rulebook you have to memorize: flip the sign here, split into two cases there, remember which direction the inequality flips.

You don't have to memorize any of that. Absolute value just means distance from zero. Once that clicks, every rule in this post falls out on its own.

What does |x| actually mean?

Forget the formal definition for a second. $|x|$ is just: how far is $x$ from $0$, on the number line, ignoring direction.

$5$ is 5 steps from zero. So is $-5$. Neither one is "more far" than the other, distance doesn't care which side you're on. That's the whole idea. We write it as:

That piecewise definition looks intimidating, but it's just saying "if you're already positive, stay put, and if you're negative, flip to your positive mirror image." Same distance, opposite side.

Now stretch this idea. $|x - a|$ isn't the distance from zero anymore, it's the distance from $a$. Drag the point below and watch the number update. It's always just the gap between the two dots, and it never goes negative.

Notice you can put $x$ on either side of $a$ and get the same reading. That's the entire reason absolute value equations tend to have two solutions: two different points can sit at the same distance from a target.

Why does |x - 1| = b give you two answers?

Let's make this concrete. Take $|x - 1| = 3$. In plain English: "which numbers are exactly 3 away from 1?"

Walk 3 steps right from 1, you land on 4. Walk 3 steps left, you land on $-2$. Both are valid, both are exactly 3 away. So:

That "or" is the two-case rule you've probably seen written as a formula. It isn't a special algebra trick, it's just the fact that distance doesn't know about direction, so both directions count.

Here's the same idea as a graph. The V shape is $y = |x - 1|$, its lowest point sits at $x = 1$ where the distance is zero. The dashed line is $y = b$, whatever target distance you pick. Wherever the line crosses the V, that's a solution.

Slide $b$ down to zero and the two crossings merge into one, there's only one point at distance zero: $x = 1$ itself. Push $b$ negative and the line never touches the V at all. A distance can't be negative, so the equation has no solution. That's not a rule to memorize, it's just geometry: the V never dips below the x-axis.

Inequalities: when "less than" means between, and "greater than" means outside

Equations ask "which points are exactly this far away?" Inequalities ask "which points are closer than this" or "farther than this." That single shift changes everything about how you read the answer.

Take $|x - 2| < 4$. This says: "I want every point within 4 units of 2." Picture standing at 2 and drawing a leash of length 4 in both directions. Everything the leash can reach is your answer, one connected stretch of the number line:

Now flip the sign: $|x - 2| > 4$. This says "farther than 4 units from 2." That's everything outside the leash's reach, two separate pieces, one on each side:

Same numbers, opposite shape of answer. "Less than" traps you in the middle. "Greater than" pushes you to both edges. Toggle between the two below and watch the shaded region flip from one solid block to two separate rays.

That's really the only thing to remember here: less-than distance gives you one interval joined by AND, greater-than distance gives you two rays joined by OR. Everything else is just solving the resulting linear inequality, which you already know how to do.

But what if it's not just x, it's a whole expression?

Everything so far used $|x - a|$, a nice clean linear expression. But absolute value bars don't care what's inside them, they'll wrap around anything: $|2x + 5|$, $|3 - x|$, whatever. Does the two-case rule still work?

Yes, and here's why. No matter what's inside the bars, that expression is either positive or negative at any given $x$. Absolute value just flips the negative parts up to make them positive, exactly like creasing a piece of paper along the spot where the expression equals zero.

Watch the fold happen. Before folding, it's a straight line, some of it above the axis, some below. After folding, everything below has been mirrored up, and you get the familiar V shape. The crease always happens exactly where the inside expression hits zero, because that's the only point where "positive" and "negative" meet.

So the rule generalizes cleanly. For $|\text{expression}| = b$:

Set up both cases, solve each one as a normal linear equation, done. The "expression" can be as messy as you like, the splitting step never changes.

The short version

Absolute value bars mean distance, and distance is always non-negative. That single fact explains everything in this post:

• $|x - a| = b$ splits into two cases because two points, one on each side of $a$, can sit at the same distance.
• If $b$ is negative, there's no solution, because distance can't be negative.
• $|x - a| < b$ gives one interval (AND), because "closer than" traps you between two boundary points.
• $|x - a| > b$ gives two rays (OR), because "farther than" pushes you past both boundaries in either direction.
• Any linear expression inside the bars folds at its own zero, so the same two-case splitting always applies, no matter how complicated the expression looks.

Next time you see absolute value bars, don't reach for a memorized rule. Just ask: how far, from what, and which directions get you there.

All visualizations are interactive React components running entirely in your browser, built with plain SVG. No libraries beyond React.